You must enter password there if you want it to be saved:
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I do following

1. startup WinSCP
   
2. select a Stored Session
   
3. Click Login button - a screen opens with header "Password", message "Prompting for credentials..."
   
4. Type in the password on the Password field
   
5. click OK button - then the server connection establishes well showing client and server folders.
   
6. Then I continue as previously described:
   I go in menu Session > Save Session - the Save Session As... screen opens
   and the checkbox "Save password (protected by master password)" is disabled
   

Did I miss the login dialog?
Apparently it does not work if you Save As a previously stored session. I don't see a reason why we should not be able to do it...
When I create a new session and then save as, then the checkbox is enabled...

It is disabled, when you have not specified any password on the Login dialog.

--> How can I save a session and store the password with it?
I have set a master password in Options > Preferences > Security.
When I do Session > Save Session, the checkbox "Save password (protected by master password)" is disabled. Am I missing something?